20th Century Philosophy of Mathematics

Carnap and Quine first met in the 1932-33 academic year, when the latter, fresh out of graduate school, visited the key centers of mathematical logic in Europe. In the months that Carnap was finishing his Logische Syntax der Sprache, Quine spent five weeks in Prague, where they discussed the manuscript “as it issued from Ina Carnap’s typewriter”. The philosophical friendship that emerged in these weeks would have a tremendous impact on the course of analytic philosophy. Not only did the meetings (...) effectively turn Quine into Carnap's disciple, they also paved the way for their seminal debates about meaning, language, and ontology---the very discussions that would change the course of analytic philosophy in the decades after the Second World War. -/- Yet surprisingly little is known about these first meetings. Although Quine has often acknowledged the impact of his Prague visit, there appears to be little information about these first encounters, except for the fact that the Quines “were overwhelmed by the kindness of the Carnaps” and that it was Quine’s “most notable experience of being intellectually fired by a living teacher”. Neither their correspondence nor their autobiographies offer a detailed account of these meetings. In this paper, I shed new light on Carnap’s and Quine’s first encounters by examining a set of previously unexplored material from their personal and academic archives. Why did Quine decide to visit Carnap? What did they discuss? And in what ways did the meetings affect Quine’s philosophical development? In what follows, I address these questions by means of a detailed reconstruction of Quine’s year in Europe based on a range of letters, notes, and reports from the early 1930s. (shrink)

One is occasionally reminded of Foucault's proclamation in a 1970 interview that "perhaps, one day this century will be known as Deleuzian." Less often is one compelled to update and restart with a supplementary counter-proclamation of the mathematician, David Lindley: "the twenty-first century would be a Bayesian era..." The verb tenses of both are conspicuous. // To critically attend to what is today often feared and demonized, but also revered, deployed, and commonly referred to as algorithm(s), one cannot avoid the (...) mathematical and philosophical legacies of probability. // But attending to these probabilistic or Bayesian legacies must include an undeniable theological legacy in which they remain entangled. // We are not, today, discovering quirky theological metaphors in contemporary technics. It's the other way around. The technologies are mere metaphors of past theologies. (shrink)

This paper describes both an exegetical puzzle that lies at the heart of Frege’s writings—how to reconcile his logicism with his definitions and claims about his definitions—and two interpretations that try to resolve that puzzle, what I call the “explicative interpretation” and the “analysis interpretation.” This paper defends the explicative interpretation primarily by criticizing the most careful and sophisticated defenses of the analysis interpretation, those given my Michael Dummett and Patricia Blanchette. Specifically, I argue that Frege’s text either are inconsistent (...) with the analysis interpretation or do not support it. I also defend the explicative interpretation from the recent charge that it cannot make sense of Frege’s logicism. While I do not provide the explicative interpretation’s full solution to the puzzle, I show that its main competitor is seriously problematic. (shrink)

In the Foundations of Mathematics, Ramsey attempted to amend Principia Mathematica’s logicism to meet serious objections raised against it. While Ramsey’s paper is well known, some questions concerning Ramsey’s motivations to write it and its reception still remain. This paper considers these questions afresh. First, an account is provided for why Ramsey decided to work on his paper instead of simply accepting Wittgenstein’s account of mathematics as presented in the Tractatus. Secondly, evidence is given supporting that Wittgenstein was not moved (...) by Ramsey’s objection against the Tractarian account of arithmetic, and a suggestion is made to explain why Wittgenstein reconsidered Ramsey’s account in the early thirties on several occasions. Finally, a reading is formulated to understand the basis on which Wittgenstein argues against Ramsey’s definition of identity in his 1927 letter to Ramsey. (shrink)